Coincidence of invariant measure for the alternate base transformations

TL;DR

This paper characterizes when two alternate base transformations share the same absolutely continuous invariant measure, focusing on the parameters defining each transformation and their associated maps.

Contribution

It provides a complete characterization of parameter pairs for which the invariant measures of two specific alternate base transformations coincide.

Findings

Identifies conditions for invariant measure coincidence

Provides explicit parameter relationships for the transformations

Enhances understanding of invariant measures in dynamical systems

Abstract

We characterize all pairs $(\beta,n),(\beta^\prime,m)$ such that the alternate $(\beta,n)$ and $(\beta^\prime,m)$-transformations $K_{(\beta,n)}$ and $K_{(\beta^\prime,m)}$ have the same absolutely continuous invariant measure, where $K_{(\beta,n)}(i,x)=(i+1 \mod 2 ,T_i(x))$ with $i\in\{0,1\}$, $T_0(x)=T_\beta (x)=\beta x \mod 1$, $T_1(x)=T_n(x)=nx\mod 1$ with $\beta>1$ real and $n\geq 2$ an integer.